The Laplace transform of $$f\left( t \right) = 2\sqrt {t/\pi } $$$$\,\,\,\,\,$$ is$$\,\,\,\,\,$$ $${s^{ - 3/2}}.$$ The Laplace transform of $$g\left( t \right) = \sqrt {1/\pi t} $$ is

We have a set of $$3$$ linear equations in $$3$$ unknown. $$'X \equiv Y'$$ means $$X$$ and $$Y$$ are equivalent statements and $$'X \ne Y'$$ means $$X$$ and $$y$$ are not equivalent statements.

$$P:$$ There is a unique solution.
$$Q:$$ The equations are linearly independent .
$$R:$$ All eigen values of the coefficient matrix are non zero .
$$S:$$ The determinant of the coefficient matrix is non-zero .

Which one of the following is TRUE?

The volume enclosed by the surface $$f\left( {x,y} \right) = {e^x}$$ over the triangle bounded by the lines $$x=y;$$ $$x=0;$$ $$y=1$$ in the $$xy$$ plane is ________.

Match the following.

List-$${\rm I}$$
$$P.$$ Stoke's Theorem
$$Q.$$ Gauss's Theorem
$$R.$$ Divergence Theorem
$$S.$$ Cauchy's Integral Theorem

List-$${\rm I}{\rm I}$$
$$1.$$
$$2.$$
$$3.$$
$$4.$$